The main result is presented in Theorem 2. According to the definition of the Fiedler vector, we have ( L + L)( f + f) = ( λ + λ)( f + f). We outline the proof below for interested readers. The main result is presented in Theorem 2. We first present Stewart's theorem in Lemma 1 to assist Actual times may differ depending on hardware and environment. We also show the number of model parameters required for each method in Table S3. Hyper-parameters were selected based on a coarse search on the validation set.

Contrastive learning methods can be applied to deep regression by enforcing label distance relationships in feature space. However, these methods are limited to labeled data only unlike for classification, where unlabeled data can be used for contrastive pretraining. In this work, we extend contrastive regression methods to allow unlabeled data to be used in a semi-supervised setting, thereby reducing the reliance on manual annotations. We observe that the feature similarity matrix between unlabeled samples still reflect inter-sample relationships, and that an accurate ordinal relationship can be recovered through spectral seriation algorithms if the level of error is within certain bounds. By using the recovered ordinal relationship for contrastive learning on unlabeled samples, we can allow more data to be used for feature representation learning, thereby achieve more robust results. The ordinal rankings can also be used to supervise predictions on unlabeled samples, which can serve as an additional training signal. We provide theoretical guarantees and empirical support through experiments on different datasets, demonstrating that our method can surpass existing state-of-the-art semi-supervised deep regression methods. To the best of our knowledge, this work is the first to explore using unlabeled data to perform contrastive learning for regression.

Neural scaling laws--power-law relationships between generalization errors and characteristics of deep learning models--are vital tools for developing reliable models while managing limited resources. Although the success of large language models highlights the importance of these laws, their application to deep regression models remains largely unexplored. Here, we empirically investigate neural scaling laws in deep regression using a parameter estimation model for twisted van der Waals magnets. We observe power-law relationships between the loss and both training dataset size and model capacity across a wide range of values, employing various architectures--including fully connected networks, residual networks, and vision transformers. Furthermore, the scaling exponents governing these relationships range from 1 to 2, with specific values depending on the regressed parameters and model details. The consistent scaling behaviors and their large scaling exponents suggest that the performance of deep regression models can improve substantially with increasing data size.

However, these methods are limited to labeled data only unlike for classification, where unlabeled data can be used for contrastive pretraining.

Contrastive learning methods can be applied to deep regression by enforcing label distance relationships in feature space. However, these methods are limited to labeled data only unlike for classification, where unlabeled data can be used for contrastive pretraining. In this work, we extend contrastive regression methods to allow unlabeled data to be used in a semi-supervised setting, thereby reducing the reliance on manual annotations. We observe that the feature similarity matrix between unlabeled samples still reflect inter-sample relationships, and that an accurate ordinal relationship can be recovered through spectral seriation algorithms if the level of error is within certain bounds. By using the recovered ordinal relationship for contrastive learning on unlabeled samples, we can allow more data to be used for feature representation learning, thereby achieve more robust results.

In deep regression, capturing the relationship among continuous labels in feature space is a fundamental challenge that has attracted increasing interest. Addressing this issue can prevent models from converging to suboptimal solutions across various regression tasks, leading to improved performance, especially for imbalanced regression and under limited sample sizes. However, existing approaches often rely on order-aware representation learning or distance-based weighting. In this paper, we hypothesize a linear negative correlation between label distances and representation similarities in regression tasks. To implement this, we propose an angle-compensated contrastive regularizer for deep regression, which adjusts the cosine distance between anchor and negative samples within the contrastive learning framework. Our method offers a plug-and-play compatible solution that extends most existing contrastive learning methods for regression tasks. Extensive experiments and theoretical analysis demonstrate that our proposed angle-compensated contrastive regularizer not only achieves competitive regression performance but also excels in data efficiency and effectiveness on imbalanced datasets.

"Just Accepted" papers have undergone full peer review and have been accepted for publication in Radiology: Artificial Intelligence. This article will undergo copyediting, layout, and proof review before it is published in its final version. Please note that during production of the final copyedited article, errors may be discovered which could affect the content. UK Biobank (UKB) has recruited more than half a million volunteers from the United Kingdom (UK), collecting health-related information on genetics, lifestyle, blood biochemistry, and more. Ongoing medical imaging of 100,000 participants, with 70,000 follow-up sessions, will furthermore yield up to 170,000 MRIs, enabling image analysis of body composition, organs, and muscle. This work presents an experimental inference engine for automated analysis of 1.5T UKB neck-to-knee body MRI.

We propose a framework for the assessment of uncertainty quantification in deep regression. The framework is based on regression problems where the regression function is a linear combination of nonlinear functions. Basically, any level of complexity can be realized through the choice of the nonlinear functions and the dimensionality of their domain. Results of an uncertainty quantification for deep regression are compared against those obtained by a statistical reference method. The reference method utilizes knowledge of the underlying nonlinear functions and is based on a Bayesian linear regression using a reference prior. Reliability of uncertainty quantification is assessed in terms of coverage probabilities, and accuracy through the size of calculated uncertainties. We illustrate the proposed framework by applying it to current approaches for uncertainty quantification in deep regression. The flexibility, together with the availability of a reference solution, makes the framework suitable for defining benchmark sets for uncertainty quantification.